Generalized coinvariant algebras for $G(r,1,n)$ in the Stanley-Reisner setting

Abstract

Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n\times n$ monomial matrices whose entries are $r^{\text{th}}$ roots of unity and let $0\le k\le n$ be an integer. Recently, Haglund, Rhoades and Shimozono ($r=1$) and Chan and Rhoades ($r>1$) introduced quotients $R_{n,k}$ (for $r>1$) and $S_{n,k}$ (for $r\ge 1$) of the polynomial ring $\mathbb{C}[x_1,\dots ,x_n]$ in $n$ variables, which for $k=n$ reduce to the classical coinvariant algebra attached to $G_n$. When $n=k$ and $r=1$, Garsia and Stanton exhibited a quotient of $\mathbb{C}[{\mathbf{y}}_S]$ isomorphic to the coinvariant algebra, where $\mathbb{C}[{\mathbf{y}}_S]$ is the polynomial ring in $2^n-1$ variables whose variables are indexed by nonempty subsets $S\subseteq [n]$. In this paper, we will define analogous quotients that are isomorphic to $R_{n,k}$ and $S_{n,k}$.